A note on multiplicity of solutions near resonance of semilinear elliptic equations

Abstract

In this paper we are concerned with the multiplicity of solutions near resonance for the following nonlinear equation: \begin{document}$ -\Delta u = \lambda u+f(x,u) $\end{document} associated with the Dirichlet boundary condition, where \begin{document}$ f $\end{document} satisfies some appropriate conditions. We will treat this problem in the framework of dynamical systems. It will be shown that there exist a one-sided neighborhood \begin{document}$ \Lambda_- $\end{document} of the eigenvalue \begin{document}$ \mu_k $\end{document} of the Laplacian operator and a dense subset \begin{document}$ {\mathcal D} $\end{document} of \begin{document}$ \mathbb{R} $\end{document} such that the equation has at least four distinct nontrivial solutions generically for \begin{document}$ \lambda\in\Lambda_- \cap {\mathcal D} $\end{document} .